Once we calculate an sample variogram, it is standard practice to smooth the empirical semivariogram by fitting a parametric model to it.
The nugget, the sill and the range parameters are often used to describe variograms:
- nugget: The random error process indicated by the height of
the jump of the semivariogram at the discontinuity at the origin.
- sill: The limit of the variogram tending to infinity lag
distances.
- range: The distance in which the difference of the variogram from the sill becomes negligible. In models with a fixed sill, it is the distance at which this is first reached; for models with an asymptotic sill, it is conventionally taken to be the distance when the semivariance first reaches 95% of the sill.
The following five parametric models are the most commonly used (Gelfand at al. 2010):
- Spherical model
\[ \gamma(h, \theta) = \begin{cases} \theta_1 (\frac{3h}{2\theta_2}- \frac{h^3}{2\theta_2^3}) & \text{for } 0\le h \le \theta_2 \\ \theta_1 & \text{for } h > \theta_2 \end{cases}\]
- Exponential model
\[\gamma(h, \theta)= \theta_1\{1-
\exp(-h/\theta_2)\}\]
- Gaussian model
\[\gamma(h, \theta)= \theta_1\{1- \exp(-h^2/\theta_2^2)\}\]
- Matérn model
\[\gamma(h, \theta)= \theta_1\left( 1- \frac{(h/\theta_2)^\nu\kappa_\nu(h/\theta_2))}{2^{\nu-1}\Gamma(\nu)}\right)\]
where \(\kappa_\nu(\cdot)\) is the modified Bessel function of the second kind of order \(\nu\).
- Power model
\[\gamma(h, \theta)= \theta_1h^{\theta_2}\]
Citation
The E-Learning project SOGA-R was developed at the Department of Earth Sciences by Kai Hartmann, Joachim Krois and Annette Rudolph. You can reach us via mail by soga[at]zedat.fu-berlin.de.
You may use this project freely under the Creative Commons Attribution-ShareAlike 4.0 International License.
Please cite as follow: Hartmann, K., Krois, J., Rudolph, A. (2023): Statistics and Geodata Analysis using R (SOGA-R). Department of Earth Sciences, Freie Universitaet Berlin.